Graphing r = 2 + 2cos(θ)

Trigonometry · Axiom Academy

Plot points at key angles to sketch the heart-shaped curve r = 2 + 2cos(θ) Maximum radius: r = 4 at θ = 0° Cusp (point) at the origin when θ = 180° Symmetric about the horizontal (polar) axis The curve traces out completely from θ = 0° to θ = 360° Excellent work! You've successfully graphed a cardioid in polar coordinates. Here's what we learned: Polar Form: Equations like r = a + b·cos(θ) or r = a + b·sin(θ) create cardioids when a = b Strategic Plotting: Choose angles where the trigonometric function is easy to evaluate (0°, 90°, 180°, 270°) The Cusp: When r = 0, the curve passes through the origin, creating the characteristic point of the cardioid Symmetry: r = 2 + 2cos(θ) is symmetric about the polar axis (horizontal line). If it were r = 2 + 2sin(θ), it would be symmetric about θ = 90° Range of r: For this cardioid, r ranges from 0 to 4, giving it a specific size Cardioids appear in many real-world applications, from microphone pickup patterns to the shape of certain heart chambers. Understanding how to plot them builds your foundation for more complex polar curves like limaçons, rose curves, and lemniscates!

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